In this paper, we are concerned with hypersurfaces in $$\mathrm{I\!H}\times \mathrm{I\!R}$$ I H × I R with constant $$r$$ r -mean curvature, to be called $$H_r$$ H r -hypersurfaces. We construct examples of complete $$H_r$$ H r -hypersurfaces, which are invariant by parabolic screw motion or by rotation. We prove that there is a unique rotational strictly convex entire $$H_r$$ H r -graph for each value $$0<H_r\le \frac{n-r}{n}$$ 0 < H r ≤ n - r n . Also, for each value $$H_r>\frac{n-r}{n}$$ H r > n - r n , there is a unique embedded compact strictly convex rotational $$H_r$$ H r -hypersurface. By using them as barriers, we obtain some interesting geometric results, including height estimates and an Alexandrov-type Theorem. Namely, we prove that an embedded compact $$H_r$$ H r -hypersurface in $$\mathrm{I\!H}^n\times \mathrm{I\!R}$$ I H n × I R is rotational ( $$H_r>0$$ H r > 0 ).